Boundary impurities are known to dramatically alter certain bulk properties of (
1+1)-dimensional strongly correlated systems. The entanglement entropy of a zero temperature Luttinger liquid bisected by a single impurity is computed using a novel finite size scaling or bosonization scheme. For a Luttinger liquid of length
2L and UV cutoff
ϵ, the boundary impurity correction ($\ensuremath{\delta}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}$) to the logarithmic entanglement entropy (${S}_{\mathrm{e}\mathrm{n}\mathrm{t}}\ensuremath{\propto}\mathrm{ln}L/ϵ$) scales as $\ensuremath{\delta}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}\ensuremath{\sim}{y}_{r}\mathrm{ln}L/ϵ$, where
yr is the renormalized backscattering coupling constant. In this way, the entanglement entropy within a region is related to scattering through the region's boundary. In the repulsive case (
g<1), $\ensuremath{\delta}{S}_{\mathrm{i}\mathrm{m}\mathrm{p}}$ diverges (negatively) suggesting that the entropy vanishes. Our results are consistent with the recent conjecture that entanglement entropy decreases irreversibly along renormalization group flow.